CN114491831B - Non-uniform material diffusion crack J integration method based on fracture phase field method - Google Patents
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Abstract
The invention discloses a non-uniform material diffusion crack J integration method based on a fracture phase field method, which comprises the following steps: 1. introducing a phase field variable d of a fracture phase field model, and determining a non-uniform material diffusion crack J integral expression with integral region independence and a corresponding finite element discrete format; 2. establishing a finite element model containing diffusion cracks and dividing a finite element grid; 3. invoking a calculation subroutine of the fracture phase field model, and solving a nonlinear control equation of the fracture phase field model by adopting a Newton-Laplasen method; 4. according to the finite element discrete format of the integral of the diffusion crack J of the heterogeneous material, the characteristic quantity-stress intensity factor K of the diffusion crack tip is solved based on the finite element calculation result I . The method realizes the accurate solution of the characteristic quantity of the diffusion crack tip of the non-uniform material, and overcomes the defect that the traditional fracture phase field method cannot accurately describe the stress state of the diffusion crack tip.
Description
Technical Field
The invention belongs to the technical field of computer simulation, relates to a method for solving a non-uniform material fracture problem, and particularly relates to a non-uniform material diffusion crack J integration method based on a fracture phase field method.
Background
The heterogeneous materials (such as functional gradient materials, composite materials and the like) have the advantages of different component performances, have designability which is difficult to compare with that of the traditional materials, and are increasingly widely applied to the fields of high-end equipment in China such as aviation, aerospace and the like. Fracture failure of heterogeneous materials is usually the result of comprehensive destructive effects such as initial damage evolution, crack nucleation, crack propagation, etc., however, the existing fracture mechanics theory still breaks through the elbows and even has bottlenecks when describing their fracture failure behavior. Common fracture mechanics models (such as a traditional finite element method, an extended finite element method, a unit deletion method, a cohesion method and the like) adopt a description mode of discrete cracks, the characteristics of crack surfaces are represented through discontinuous displacement fields, and crack surface topology needs to be tracked in a simulation process. The methods are all based on the premise that cracks exist, and cannot describe the initial damage evolution and crack nucleation process of the material. The fracture phase field method is a new fracture mechanics model based on a continuous damage model, and adopts a dispersion area with limited width to approximately represent a discrete crack surface. The fracture phase field method is favored by more and more students because the fracture phase field method adopts a phase field scalar d to represent the damage state of the whole-domain material and all field variables are all continuous in the whole domain, and can cover a series of complex physical phenomena of crack from no to complete fracture failure of the material.
The fracture phase field method has two main characteristics: (1) Realizing that the dispersion crack characterized by the phase field order parameter d replaces the traditional discrete crack by introducing the crack surface density gamma (d, d); (2) The crack initiation and propagation evolution process is controlled by the phase field evolution equation without additional fracture criteria. The method is precisely limited by the two characteristics, and on the one hand, the fracture phase field method cannot accurately capture the position of the crack tip, so that the characteristic quantity of the crack tip is difficult to effectively extract; on the other hand, the influence mechanism of the phase field evolution equation on the propagation rule of the diffusion crack is not clear. The effective extraction of the characteristic quantity of the tip of the diffusion crack is important to the elucidation of the influence mechanism of a phase field evolution equation on the propagation rule of the diffusion crack. Therefore, the calculation method capable of effectively extracting the characteristic quantity of the tip of the diffusion crack is very important for the application and development of a fracture phase field method.
Disclosure of Invention
The invention aims to provide a non-uniform material diffusion crack J integration method based on a fracture phase field method, which derives a J integration expression of a non-uniform material diffusion crack by introducing a phase field related variable, and theoretically proves the independence of an integration area (size and shape) of the diffusion crack J integration; based on a finite element method, the numerical implementation is carried out on the integral of the diffusion crack J, and compared with the traditional fracture mechanics method, the effectiveness of the integral of the diffusion crack J is verified, so that the solution of the characteristic quantity of the diffusion crack tip is realized.
The invention aims at realizing the following technical scheme:
a non-uniform material diffusion crack J integration method based on a fracture phase field method comprises the following steps:
step one: introducing a phase field variable d of a fracture phase field model, deducing that the integral of the non-uniform material diffusion crack J meets the requirement of integral region independence, and determining an integral expression of the non-uniform material diffusion crack J with the integral region independence and a corresponding finite element discrete format, wherein:
the integral expression of the non-uniform material diffusion crack J with independence of integral areas is as follows:
the finite element discrete format for non-uniform material propagation crack J integration with integral region independence is:
wherein q is a smooth function, sigma ij As stress tensor, u i As displacement tensor, ζ j For phase field stress, ψ e Delta as strain energy ij Is a Kronecker function (where i=1),for the space coordinate x i Is used for the partial derivative of (a),a is the area surrounded by a closed curve Γ, N e For the number of units in the integration area, m is the number of integration points in each unit, |J| p Jacobian determinant, w, of a certain integral point p p I, j=1, 2, which is the weight of the integration point;
step two: establishing a finite element model containing a propagation crack by means of finite element software, dividing a finite element grid, carrying out grid refinement on a region near the tip of the propagation crack so as to meet convergence requirements of finite element calculation, and defining boundary conditions, load information, material attribute information and the like;
step three: aiming at the finite element model with the dispersion cracks built in the second step, invoking a calculation subprogram of the fracture phase field model through finite element software, solving a nonlinear control equation of the fracture phase field model by adopting a Newton-Laplasen method, and outputting information such as load, displacement, stress, strain, phase field distribution and the like;
step four: according to the finite element discrete format of the non-uniform material diffusion crack J integral of the first step, writing a calculation program of the non-uniform material diffusion crack J integral; solving characteristic quantity-stress intensity factor K of diffusion crack tip based on finite element calculation result in step three I Wherein:
stress intensity factor K I The calculation formula of (2) is as follows:
wherein,,j integral, E for non-uniform material propagation crack tip And v tip Young's modulus and Poisson's ratio at the crack tip, respectively.
Compared with the prior art, the invention has the following advantages:
1. the invention introduces the phase field variable d of the fracture phase field model, and solves the characteristic quantity (stress intensity factor K) of the non-uniform material diffusion crack tip through constructing the regional conservation integral of the diffusion crack tip I ). Compared with the traditional method, the invention still has independence of integral area for the non-uniform material, realizes the characteristic quantity (stress intensity factor K) of the diffusion crack tip of the non-uniform material I ) The defect that the traditional fracture phase field method cannot accurately describe the stress state of the tip of the diffusion crack is overcome.
2. By changing model information, boundary conditions, load information, material attribute information and the like, the invention can realize the characteristic quantity (stress intensity factor K) of the tip of the diffusion crack of different materials under different stress states I ) Is a solution to (c).
3. The method is suitable for the problem of non-uniform material fracture with complex material properties, has high calculation precision and efficiency, is developed and realized based on a general finite element calculation software platform, and has strong versatility and flexibility.
Drawings
FIG. 1 is a block flow diagram of a non-uniform material propagation crack J integration method based on a fracture phase field method;
FIG. 2 is a schematic illustration of geometry, boundary conditions, loading and material property settings for a test specimen of a tensile non-uniform material containing a single crack;
FIG. 3 is a schematic view of a finite element mesh containing a single-sided crack tensile non-uniform material specimen;
FIG. 4 is a graph of a crack propagation morphology of a non-uniform material sample simulated using a fracture phase field method;
FIG. 5 is a graph comparing the stress intensity factor solved by the integral J of the non-uniform material propagation crack with the theoretical solution and the integral J of the uniform material propagation crack.
Detailed Description
The following description of the present invention is provided with reference to the accompanying drawings, but is not limited to the following description, and any modifications or equivalent substitutions of the present invention should be included in the scope of the present invention without departing from the spirit and scope of the present invention.
As shown in FIG. 1, the invention provides a heterogeneous material diffusion crack J integration method based on a fracture phase field method, which is supported by national natural science foundation (project approval number: 12002106) and mechanical structure strength and vibration national emphasis laboratory (Western An transportation university) open subject (project approval number: SV 2021-KF-07). The method comprises the following steps:
step one: aiming at a non-uniform material diffusion crack model of a fracture phase field method, a phase field variable d is introduced, and the influence of material non-uniformity and the phase field variable on J integral conservation is quantified. The basic material properties (such as Young's modulus, poisson's ratio, etc.) of the heterogeneous material and the corresponding fracture phase field method input parameters (characteristic length parameters, fracture energy density, etc.) are functions of spatial positions, and the derivative of the spatial coordinates is generally not 0, so the influence of these factors on the J integral conservation is comprehensively considered, the J integral expression of the heterogeneous material dispersion crack with integral region independence is deduced, and the deduction result is as follows:
ψ＝ψ e +ψ f ；
wherein,,for the energy-momentum tensor of the propagating crack, Γ is any closure surrounding the propagating crack tipCurve A is the area enclosed by closed curve Γ, n j Is the external normal unit vector of a closed curve Γ, and ψ is the free energy of the system e Is strain energy, ψ f For breaking energy, delta ij As Kronecker function (i=1 in the formula), σ ij As stress tensor, u i As displacement tensor, ζ j For phase field stress>For the space coordinate x i G (d) is the energy degradation function, C ijkl (x) Epsilon is the fourth order stiffness tensor of the material ij And epsilon kl G is the strain tensor f (x) To the fracture energy density (i.e. the energy required to form a crack propagating per unit area), l c (x) Omega, as a length scale parameter d C is a geometric function ω For a scaling factor related to the geometric function, < ->Is a hamiltonian differential operator, i, j, k, l=1, 2.
Since the line integral form of J integration is disadvantageous for numerical calculations, it is generally necessary to convert it into an equivalent area integral form that can be achieved in finite elements. Based on the divergence theorem, an equivalent area integral expression of the J integral of the diffusion crack of the heterogeneous material is deduced, and the deduction result is as follows:
where q is a smooth function (1 at the crack tip and 0 at the boundary Γ of a).
To correlate the energy release rate in the phase field fracture model with the traditional line elastic fracture mechanics, the effective energy-momentum tensor is redefined
Equivalent area integration with non-uniform material having integrated area independenceCan be defined as:
based on the finite element thought, a finite element discrete format of the J integration of the diffusion crack of the heterogeneous material is deduced, and the deduction result is as follows:
wherein N is e Is the number of units in the integration area; m is the number of integration points in each unit; i J p Jacobian, which is a certain integral point p; w (w) p Is the weight of the integration point.
Step two: establishing a finite element model containing a propagation crack by means of finite element software ABAQUS and dividing a finite element grid; in order to ensure the calculation accuracy, the area near the tip of the diffusion crack is subjected to grid refinement; defining boundary conditions, load information, material attribute information and the like; an abaqus.inp file containing model information of units, nodes, boundary conditions, loads, material properties, etc. is generated.
Step three: modifying an abaqus.inp file, writing command lines for calling a fracture phase field model UEL and a UMAT subroutine subcarrier for, submitting an input file abaqus.inp and a subroutine file subcarrier for to ABAQUS software, and inputting corresponding commands for solving calculation. After the calculation is completed, the ABAQUS software checks the calculation result, outputs information such as load, displacement, stress, strain and phase field distribution, and the result file is named as data.out.
Step four: and (3) programming a calculation program of J integral of the crack propagation of the heterogeneous material by MATLAB language, and calculating the J integral of the crack propagation (namely the energy release rate of the crack propagation tip) based on the result information output by ABAQUS. The stress intensity factor is calculated as follows:
wherein E is tip And v tip Young's modulus and Poisson's ratio at the crack tip, respectively. According to the above, the stress intensity factor K of the non-uniform material diffusion crack tip under the plane stress and plane strain state can be solved I . By selecting integration areas with different sizes, the independence of the integration areas of the crack propagation J integration method is compared and verified, and the correctness of the traditional J integration calculation result is compared and verified.
To illustrate the performance of the above-described aspects of the invention, a further description is provided below in conjunction with an example.
Consider a non-uniform material fracture problem under planar strain. As shown in fig. 2, a square plate is provided with a horizontal crack on the side, the tip of the crack is positioned right in the center of the plate, and an orthogonal rectangular coordinate system is established. The width and height of the plates were w=1.0 mm and h=1.0 mm, respectively. The length of the horizontal crack a=0.5 mm. The bottom of the plate is constrained and applied along x at the top 2 Displacement load in direction u=1.0×10 -3 mm. The displacement increment is Δu=1.0×10 -5 mm. As shown in fig. 3, in order to secure the calculation accuracy, the area where the crack tip is propagated is subjected to mesh refinement, and the effective mesh size of the refined area is about 0.0005mm. The finite element model contains 20304 fully integrated quadrilateral elements, 20563 nodes, and 61689 degrees of freedom in total. With an alumina (Al 2 O 3 ) Zirconium oxide (ZrO) 2 ) The functional gradient plate of the ceramic composite material has the following material properties and input parameters: young's modulus E 1 =380 GPa, poisson ratio v 1 Fracture toughness k=0.26 IC,1 ＝5.2MPa·m 1 /2 Young's modulus E 2 =210 GPa, poisson ratio v 2 Fracture toughness k=0.31 IC,2 ＝9.6MPa·m 1/2 Characteristic length parameter l c =0.0025 mm (wherein, (. Cndot.) 1 And ( 2 Respectively represent alumina (x) 1 = -W/2) and zirconia (x 1 Material properties of =w/2)). Assuming Young's modulus E, poisson's ratio v, fracture toughness K of the functionally graded plate IC All along x 1 = -W/2 to x 1 Change exponentially on =w/2, i.e.:
for plane strain conditions, the input parameter g in the phase field fracture model f (x 1 ) Determined by the following formula:
FIG. 4 shows a graph of the morphology of a propagating crack in a sample of non-uniform material simulated using a fracture phase field method. FIG. 5 shows the propagation crack tip stress intensity factor K for different integral zone sizes I It can be seen that when the size r of the integration region is > 20l c (i.e., r > 0.05 mm), the calculation result of the non-uniform material propagation crack J integration method of the present invention does not change with the change of the integration area size. The result shows that the non-uniform material propagation crack J integration method has independence of integration areas. Furthermore, the junction of existing methods for J integration of uniform material propagation cracksThe fruit cannot be matched with the theoretical solution all the time; in contrast, when the dimension r of the integration region is > 20l c When r is more than 0.05mm, the calculation result of the non-uniform material diffusion crack J integration method has very good consistency with a theoretical solution, and the correctness of the scheme of the invention is verified by comparison.
The above examples verify the independence and correctness of the integration area of the technical scheme of the invention.
Claims (3)
1. A method for integrating a diffusion crack J of a heterogeneous material based on a fracture phase field method, which is characterized by comprising the following steps:
step one: introducing a phase field variable d of a fracture phase field model, deducing that the integral of the non-uniform material diffusion crack J meets the requirement of integral region independence, and determining a non-uniform material diffusion crack J integral expression with integral region independence and a corresponding finite element discrete format;
step two: establishing a finite element model containing a propagation crack by means of finite element software, dividing a finite element grid, carrying out grid refinement on a region near the tip of the propagation crack so as to meet convergence requirements of finite element calculation, and defining boundary conditions, load information and material attribute information;
step three: aiming at the finite element model with the dispersion cracks, which is established in the second step, solving a nonlinear control equation of a fracture phase field model by adopting a Newton-Laplasen method, and outputting load, displacement, stress, strain and phase field distribution information;
step four: according to the finite element discrete format of the diffusion crack J integration of the heterogeneous material in the first step, the characteristic quantity-stress intensity factor K of the diffusion crack tip is solved based on the finite element calculation result in the third step I 。
2. The method for integrating the diffusion cracks of the heterogeneous material based on the fracture phase field method according to claim 1, wherein in the first step, the integration expression of the diffusion cracks of the heterogeneous material with independence of integration areas is:
the finite element discrete format for non-uniform material propagation crack J integration with integral region independence is:
wherein q is a smooth function, sigma ij As stress tensor, u i As displacement tensor, ζ j Is phase field stress, d is phase field parameter, ψ e Delta as strain energy ij Is a Kronecker function and where i=1 is taken,for the space coordinate x i A is the area enclosed by a closed curve Γ, N e For the number of units in the integration area, m is the number of integration points in each unit, |J| p Jacobian determinant, w, of a certain integral point p p I, j=1, 2, which is the weight of the integration point.
3. The method for integrating diffusion cracks in heterogeneous material based on fracture phase field method according to claim 1, wherein in the fourth step, the stress intensity factor K I The calculation formula of (2) is as follows:
wherein,,is non-uniformMaterial propagation crack J integral, E tip And v tip Young's modulus and Poisson's ratio at the crack tip, respectively.
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